Then at least be upfront with the user about why you're asking, because they might well answer differently (e.g. include a different number of parts of their last name) if they know your purpose is to sort the name than if they think it's being used for a different purpose. They might even give totally different names.

That's the most important part of the comment above: The concept of a name is so overloaded that unless you ask about the string to use for the specific purposes you intend to use it, then there is very little you can do with it.